The function \(f (x)\) is defined on the positive real \(x\) and takes only positive values. It is known that \(f (1) + f (2) = 10\) and \(f(a+b) = f(a) + f(b) + 2\sqrt{f(a)f(b)}\) for any \(a\) and \(b\). Find \(f (2^{2011})\).
The numerical function \(f\) is such that for any \(x\) and \(y\) the equality \(f (x + y) = f (x) + f (y) + 80xy\) holds. Find \(f(1)\) if \(f(0.25) = 2\).
Does there exist a function \(f (x)\) defined for all real numbers such that \(f(\sin x) + f (\cos x) = \sin x\)?
Are there functions \(p (x)\) and \(q (x)\) such that \(p (x)\) is an even function and \(p (q (x))\) is an odd function (different from identically zero)?
For all real \(x\) and \(y\), the equality \(f (x^2 + y) = f (x) + f (y^2)\) holds. Find \(f(-1)\).